Amplitude modulation of a radio wave

Electromagnetic waves are widely used for the purpose of communication as in radio transmissions and mobile telephones. One method of radio transmission employs amplitude modulation (AM). Here, the amplitude of a sinusoidal electromagnetic wave, called the carrier wave, is varied to carry the required information which might be speech or music. The principle of AM is illustrated in Figure 1(a). Here, the amplitude of the carrier wave of frequency ω_c is modulated by a sinusoidal wave of much lower frequency ω_m. The resultant wave can be represented by

......(1)

B is called the depth of modulation, which must be less than A to avoid distortion of the signal at the receiver. Using the trigonometric identity

......(2)

we can rewrite Equation (1) as

.......(3)

Figure 1 (a) The principle of AM radio transmission. A carrier wave of frequency ω_c is modulated by a sinusoidal wave of frequency ω_m, where ω_m << ω_c. The resultant waveform ψ is shown for x = 0. (b) The frequency spectrum of the modulated carrier wave showing the frequency components present.

Inspection of Equation (3) shows that there are three frequency components present in the modulated wave. These are the carrier frequency ωc and the frequencies (ω_c + ω_m) and (ω_c − ω_m). We can represent these components as a frequency spectrum as shown in Figure 1(b). In this spectrum the heights of the lines represent the amplitudes of the frequency components and in this particular example, B = A/3. Of course a real audio signal contains a continuous range of frequencies, typically 10 Hz to 10 kHz, and so ωm will have this range also. (This compares with the carrier frequency which is typically ∼1 MHz, i.e. ω_c >> ω_m.)
Consequently there is a band of frequencies on either side of the central frequency ω_c, which are called side bands. It follows that adjacent radio stations must have carrier frequencies that differ by more than 2ω_m. (In more sophisticated AM transmission systems, only the frequencies of a single side band are transmitted so that more radio stations can fit into the available frequency range.)